In this section, we compare the performance of different strategies with σ2
r =σs2 and Pr =Ps = 1 in all cases.
Fig. 2.2 shows the optimized non-abs-based relay function for different SNRs and different values ofh1 and h2. The relay operation behaves like the AF strategy at low SNR and like the DF strategy (2.12) at high SNR. Different abs-based relay functions f(u) are compared in Fig. 2.3, where for AAF we choose C =√Ps+σr/
√
2. Unlike ADF with a hard limiter, the optimized relay adapts its transmit power according to the signal strength it receives; This is the benefit of the average power constraint. From Fig. 2.3, we can also see that when the SNR is small, the optimized relay function gives a “V”-shaped behavior similar to that of the AAF strategy. As the SNR increases, the behavior of the optimized relay function is more similar to the one for the ADF strategy. This suggests that ADF performs well at high SNR while AAF is effective at low SNR. Interestingly, the relay function of EF has almost the same shape as the optimized relay function at all SNRs.
Fig. 2.4 compares the bit error rate (BER) performance of different abs-based and non-abs-based strategies for BPSK when h1 = 1 and h2 = 0.8. We observe that the
−10 −8 −6 −4 −2 0 2 4 6 8 10 −2 −1 0 1 2 3 4 f(u) u AAF ADF Optimized −5 dB Optimized 0 dB Optimized 5 dB EF −5 dB EF 5 dB
Figure 2.3: Comparison of functionf(u) in different abs-based schemes withσ2 r =σs2, h1 =h2 = 1 andPr =Ps= 1.
optimized non-abs-based (abs-based) relay performs like the AF (AAF) strategy at low SNR and like the DF (ADF) strategy at high SNR. Also, EF performs close to the optimized strategy for all SNR values. It can also be seen that in this scenario non-abs-based strategies perform better than abs-based strategies at low SNR and worse than abs-based strategies at high SNR. The reason for this is that non-abs- based strategies do not exploit the a priori information about the signal available at each terminal providing; Thisa priori information produces extra redundancy, which is useful particularly at low SNR. A similar behavior is observed in Fig. 2.5, where the case h1 = 1 and h2 = 0.5 is considered. Compared to the results for h1 = 1 and h2 = 0.8 in Fig. 2.4 the threshold SNR below which non-abs-based strategies perform better than abs-based strategies is increased. Thus, non-abs-based strategies are beneficial for asymmetric channels.
In Fig. 2.6 we compare the BER for the AF, AAF, and ADF strategies on the two- way relay channel in the high SNR regime, where we assume that σ2
r =σs2, h1 = h2 and Pr = Ps = 1. For the AAF strategy we set C = 1. Also, we do not include the
−5 0 5 10 10−3 10−2 10−1 100 BER −2 −1 0 1 2 3 4 0.25 0.3 0.35 0.4
Optimized Abs Relay ADF Optimal w and v AAF Optimal v, C=h
1Ps
0.5
Abs EF Optimal v=0, C=h1Ps0.5 Optimized Non−abs Relay DF Optimal
AF Non−abs EF
SNR (dB)
Figure 2.4: Performance comparison between different abs-based and non-abs-based strategies when h1 = 1 andh2 = 0.8,Pr =Ps = 1. The subfigure shows the crossover between the abs-based and non-abs-based strategies.
−5 0 5 10 10−2 10−1 100 SNR (dB) BER 0 1 2 3 4 5 6 0.25 0.3 0.35 0.4
Optimized Abs Relay ADF Optimal w and v AAF Optimal v, C=h1Ps0.5 Abs EF Optimal v=0, C=h1Ps0.5 Optimized Non−abs Relay DF Optimal
AF Non−abs EF
Figure 2.5: Performance comparison between different abs-based and non-abs-based strategies when h1 = 1 andh2 = 0.5,Pr =Ps = 1. The subfigure shows the crossover between the abs-based and non-abs-based strategies.
optimized relay and EF strategies as their performances are very close to ADF at high SNR. We observe from Fig. 2.6 that AAF has a 2 dB gain over AF at a BER of 10−8. Finally, we can see from Fig. 2.6 that ADF performs best, where a 2.7 dB gain over AAF at a BER of 10−8 can be observed.
In Fig. 2.7 the average error probability of ADF and DF is compared for three different cases. The plotted results agree with our analysis in Section 2.4. Fig. 2.8 compares the behavior off(u) for AAF, ADF, and EF strategies for σ2
r =σs2, where both terminals use 4-PAM, i.e., M = |V′| = 4,5,6,7, and the SNR is chosen to be
5/σ2
r. The behavior of the relay function in Fig. 2.8 resembles the one in Fig. 2.3 for different strategies. In particular, EF and AAF perform similarly when the SNR is low. As the SNR increases, the EF relay function gives performance resemblig the behavior of the ADF relay function.
In Fig. 2.9 the symbol error rate (SER) of different relay functions using ADF and AAF is compared, where the same parameters as in Fig. 2.8 are used. The
5 10 15 20 10−6 10−5 10−4 10−3 10−2 10−1 100 SNR (dB) BER AF AAF ADF
Figure 2.6: Performance comparison of AF, AAF and ADF with the AF scheme for one-way relay channel when σ2
r =σ2s,h1 =h2 and Pr=Ps = 1. 0 2 4 6 8 10 12 14 16 18 20 10−5 10−4 10−3 10−2 10−1 100 SNR (dB) Average BER DF, h 1=1, h2=1, Pr=0.5 ADF, h 1=1, h2=1, Pr=0.5 DF, h 1=1, h2=0.2, Pr=2 ADF, h1=1, h2=0.2, Pr=2 DF, h 1=1, h2=0.8, Pr=5 ADF, h1=1, h2=0.8, Pr=5
−10 −8 −6 −4 −2 0 2 4 6 8 10 −6 −4 −2 0 2 4 6 u f(u) EF SNR=5 dB EF SNR=15 dB AAF ADF
Figure 2.8: Comparison of relay functions for AAF, ADF, and EF withσ2
r =σ2s where both terminals use 4-PAM.
0 5 10 15 20 10−4 10−3 10−2 10−1 100 SNR (dB) SER ADF, M=4, (50) ADF, M=4, (51) ADF, M=5 ADF, M=6 DF AAF, M=4, (52) AAF, M=5 AAF, M=6 AF 2 4 6 8 10 0.2 0.3 0.4 0.5 0.6 0.7
Figure 2.9: SER comparison of ADF and AAF relay functions for 4-PAM with M =
|V′| = 4,5,6,7 and σ2
r = σ2s. The subfigure shows the crossover between different strategies.
performance degrades asMincreases. In Fig. 2.9, a comparison between the mappings in (2.50) and (2.51) shows almost identical performance. There are two factors that affect the performance of relay functions with differentM. First, a small M indicates a higher compression at the relay, which results in power savings. Second, whenM is small, a detection error at the relay may affect the overall performance. At high SNR, it is clear that the power savings dominate the performance of ADF. At low SNR, we find that the performance degrades asM decreases, which means thatM = 7 achieves the best performance. For example, at SNR= 0 dB, the SERs forM = 4, 5, 6, 7 are 0.6904, 0.6472, 0.6428, and 0.6146, respectively. This observation generalizes the one for the BPSK case, where the reason for this behavior is again that the redundancy in the constellation set increases for larger M.